3 Answers
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I'm not sure how it works in the GOTY edition, but if you purchase the expansion packs separately then the missions will appear in your quest log automatically (after a text box pop-up describing some radio signal or something similar).

The only expansion which ties into the main story is Broken Steel - it is an extension of the main storyline beyond the regular ending, and allows you to continue playing after completing the main quest line (unlike the original).

The others are all totally standalone, all taking place in alternate locations to the Capital Wasteland, and so it makes no difference to the main story when you start them.

In terms of toughness - Point Lookout definitely has the hardest enemies to fight, so I would tackle that one later on. The others are fairly playable at any decent level I think.

Here there be spoilers

I recommend doing Operation Anchorage as early as you reasonably can. I just redid it, and realized that one of the rewards for completion of that pack is power armor training -- so if you do it early enough, you'll be able to wear power armor way before you normally would in the game. (Unless you were fast-tracking the storyline missions. But who does that? :)

For the others, there are definite advantages to doing some early (Mothership Zeta comes to mind, as some of the new items from it are really powerful), but as C.McAtackney said, the enemies can prove more challenging in some areas (notably Point Lookout).

On the plus side, now that you're playing with all the add-ons, you'll be able to go up to Level 30 ... And that's regardless of whether or not you even make it to the Broken Steel missions. :) The way I'm currently replaying, I'm liable to hit level 30 before I find Dad ... :)

Like the other guys said, in some of the DLC's the enemies will be more challenging. If you start Operation Anchorage early, you can train power armor, get a strong suit of armor, and a couple of weapons with lots of firepower.

With Mothership Zeta you get one of two of the strongest weapons in the game.

Do either those two first, and you'll be prepared for most challenges you face in the game, especially Point Lookout.